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Transform an image under an arbitrary projection? Looks like a job for ImageTransformation :)
@halirutan's cart function gives you a mapping from latitude and longitude to the Mollweide projection. What we need here is the inverse mapping, because ImageTransformation is going to look at each pixel in the Mollweide projection and fill it in with the colour ...

Update June 2015
Here is an updated version of the program. I've made it compatible with newer Mathematica versions (AstronomicalData returns Quantity structures in newer versions, which wrangled calculations). It should now work on versions 8 through 10. Let me know if it doesn't.
I added animation and simplified the presentation (no tooltips in the ...

Yes, the basic idea is here: Demonstration: Day and Night World Clock
Now, to use the images, create an alpha channel using the computed the day-night curve--called "terminator" curve (rasterize it in grayscale), and compose two images using ImageCompose with the generated alpha channels (SetAlphaChannel to the second image).
Try the following code:
a = ...

Stars
RA and Dec for stars can be fetched via
StarData["Sirius", {"RightAscension", "Declination"}]
(* -> {6 h, 45 m, 9.3 s, -16 degrees, -42 arc minutes, -47.2 arc seconds} *)
Although one can specify a particular date and time for these coordinates, the result Mathematica gives does not actually depend on the date or time at all - an indication ...

It took me quite a while, but finally, here's a visualization of the perigee of Flamsteed's comet:
I should first note two things: first, some of the needed data for computing the orbit of comet C/1683 O1 was missing in AstronomicalData["CometC1683O1", "Properties"], and I had to pull information from external sources to supplement the information ...

Offered as an alternative to getting the same information and a check on it, one can also get this measurement from a WolframAlpha query:
...
Of some interest, by these measurements Jupiter appears to have moved quite a ways further from the Sun since belisarius's answer just some 11 hours ago.
67.74204*10^11 vs 7.74232*10^11
WolframAlpha can also ...

data = Cases[ Import[FileNames["*.dat"][[1]]],
{a_, b_, c_} :> {b, Mod[a, 360, -180]}]; (*thanks to bbgodfrey*)
To show points you have to stick with GeoGraphics. GeoListPlot is designed for Entities.
To add something more to the question I changed Ra to hours.
GeoGraphics[{Red, Point@GeoPosition@data},
GeoRange -> {All, ...

Well I decided to give it a bit of a go...First import the image and convert to grayscale, then crop to focus on the area of interest. Then I used a LaplacianGaussianFilter, which is often used in blob detection.
img = ImageAdjust@ColorConvert[Import["http://i.imgur.com/4lDwE33.jpg"], "Grayscale"];
smallimg = ImageAdjust@ImageTake[img, {200, 500}, {200, ...

Jean Meeus's Astronomical Algorithms (as well as the related book Astronomical Formulæ for Calculators) is what you should start looking at whenever you need to deal with algorithms for quantities of astronomical interest.
For instance, here is a translation of Meeus's method for the Julian Date:
Options[jd] = {"Calendar" -> "Gregorian"};
...

This is simplest implementation. If a new crater gets closer than 30 to some old craters, only closest old crater is getting replaced with new one. You can built on this example something more sophisticated.
craters = {{0, 0}};
number = {1};
Dynamic[new = RandomReal[{-250, 250}, 2];
near = Nearest[craters, new][[1]];
Row[{
Graphics[{PointSize[.05],
...

You have most of the pieces here already.
UPDATED
The proper coordinates
In this problem, we want to get the positions of astronomical objects in terms of a Cartesian system that is geocentric and rotates with the Earth. In astronomy, the positions of objects are commonly given in terms of right ascension (RA) and declination (Dec), which are similar to ...

A couple of years ago I was in an email conversation about this topic with Jeff Bryant, a WRI employee. He was not directly responsible for AstronomicalData, but he told me that Mathematica did not correct for atmospheric refraction. Good to know, as the refraction at the horizon is about the same size as the sun itself (both in degrees).
At that time I ...

This is not a full answer, but more a response to J.M.'s comment and provides a routine to calculate $\Delta T$ which was sitting on my hard disk. This is intended as a starting point for further calculations.
deltaT::usage =
"deltaT[date] calculate the arithmetic difference, in seconds, \
between the Terrestrial Dynamical Time (TD) and the Universal \
...

Search for brfASTRO.m which is a fantastic astronomy package that Peter Breitfeld wrote. He offers wonderful material on his homepage. I used many of his routines (thank you Peter!)
Here is my (long but untested) version which I probably also copied partly and forgot from where. Please contact me if you feel that proper credit is due!
This is a small part ...

A bit of spelunking reveals that AstronomicalData delegates the calculation of those properties to the function PlanetaryAstronomy`Private`RiseSetsX. You can verify this by evaluating:
On[PlanetaryAstronomy`Private`RiseSetsX]
AstronomicalData["Moon", "NextRiseTime"]
Off[PlanetaryAstronomy`Private`RiseSetsX]
My impression is that it is performing a purely ...

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